SCHWEITZER 2020 Problem 8

Let $\mathbb{F}_{p}$ denote the $p$ -element field for a prime $p>3$ and let $S$ be the set of functions from $\mathbb{F}_{p}$ to $\mathbb{F}_{p}$ . Find all functions $D\colon S\to S$ satisfying $$D(f\circ g)=(D(f)\circ g)\cdot D(g)$$ for all $f,g \in {S}$ . Here, $\circ$ denotes the function composition, so $(f\circ g)(x)$ is the function $f(g(x))$ , and $\cdot$ denotes multiplication, so $(f\cdot g)=f(x)g(x)$ .